Observe that, the right hand part in the preceding inequality is free from \(y\) . Thus, using the continuity of \(\Psi \) , \(|t_2-t_1|\rightarrow 0 \)  implies that \(|\mathcal {A}y(t_2)-\mathcal {A}y(t_1)|\rightarrow 0\) . This proves that \(\mathcal {A}(\mathcal {U})\)  is equicontinuous. In the view of Arzela-Ascoli Theorem [1]}, it follows that \(\mathcal {A}(\mathcal {U})\)  is relatively compact. Therefore, by Schauder's fixed point Theorem REF , operator \(\mathcal {A}\)  has at least one fixed point \(y \in C_{1-\zeta \,; \,\Psi }\left[ a,a+\chi \right] \)  which is the solution of IVP (REF )–().
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